Find the x and y Intercepts f(x)=-4x^2-10x-2

$f (x) = - 4 x^{2} - 10 x - 2$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 4 x^{2} - 10 x - 2$

Solve the equation.

Rewrite the equation as $- 4 x^{2} - 10 x - 2 = 0$ .

$- 4 x^{2} - 10 x - 2 = 0$

Factor $- 2$ out of $- 4 x^{2} - 10 x - 2$ .

Factor $- 2$ out of $- 4 x^{2}$ .

$- 2 (2 x^{2}) - 10 x - 2 = 0$

Factor $- 2$ out of $- 10 x$ .

$- 2 (2 x^{2}) - 2 (5 x) - 2 = 0$

Factor $- 2$ out of $- 2$ .

$- 2 (2 x^{2}) - 2 (5 x) - 2 \cdot 1 = 0$

Factor $- 2$ out of $- 2 (2 x^{2}) - 2 (5 x)$ .

$- 2 (2 x^{2} + 5 x) - 2 \cdot 1 = 0$

Factor $- 2$ out of $- 2 (2 x^{2} + 5 x) - 2 (1)$ .

$- 2 (2 x^{2} + 5 x + 1) = 0$

Divide each term in $- 2 (2 x^{2} + 5 x + 1) = 0$ by $- 2$ and simplify.

Divide each term in $- 2 (2 x^{2} + 5 x + 1) = 0$ by $- 2$ .

$\frac{- 2 (2 x^{2} + 5 x + 1)}{- 2} = \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 (2 x^{2} + 5 x + 1)}{- 2} = \frac{0}{- 2}$

Divide $2 x^{2} + 5 x + 1$ by $1$ .

$2 x^{2} + 5 x + 1 = \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

$2 x^{2} + 5 x + 1 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 2$ , $b = 5$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (2 \cdot 1)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 8}}{2 \cdot 2}$

Subtract $8$ from $25$ .

$x = \frac{- 5 \pm \sqrt{17}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm \sqrt{17}}{4}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 8}}{2 \cdot 2}$

Subtract $8$ from $25$ .

$x = \frac{- 5 \pm \sqrt{17}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm \sqrt{17}}{4}$

Change the $\pm$ to $+$ .

$x = \frac{- 5 + \sqrt{17}}{4}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + \sqrt{17}}{4}$

Factor $- 1$ out of $\sqrt{17}$ .

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{17})}{4}$

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{17})$ .

$x = \frac{- 1 (5 - \sqrt{17})}{4}$

Move the negative in front of the fraction.

$x = - \frac{5 - \sqrt{17}}{4}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot 1}}{2 \cdot 2}$

Multiply $- 8$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 8}}{2 \cdot 2}$

Subtract $8$ from $25$ .

$x = \frac{- 5 \pm \sqrt{17}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm \sqrt{17}}{4}$

Change the $\pm$ to $-$ .

$x = \frac{- 5 - \sqrt{17}}{4}$

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - \sqrt{17}}{4}$

Factor $- 1$ out of $- \sqrt{17}$ .

$x = \frac{- 1 \cdot 5 - (\sqrt{17})}{4}$

Factor $- 1$ out of $- 1 (5) - (\sqrt{17})$ .

$x = \frac{- 1 (5 + \sqrt{17})}{4}$

Move the negative in front of the fraction.

$x = - \frac{5 + \sqrt{17}}{4}$

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{17}}{4}, - \frac{5 + \sqrt{17}}{4}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{5 - \sqrt{17}}{4}, 0), (- \frac{5 + \sqrt{17}}{4}, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 4 {(0)}^{2} - 10 \cdot 0 - 2$

Solve the equation.

Remove parentheses.

$y = - 4 \cdot 0^{2} - 10 \cdot 0 - 2$

Remove parentheses.

$y = - 4 {(0)}^{2} - 10 \cdot 0 - 2$

Simplify $- 4 {(0)}^{2} - 10 \cdot 0 - 2$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = - 4 \cdot 0 - 10 \cdot 0 - 2$

Multiply $- 4$ by $0$ .

$y = 0 - 10 \cdot 0 - 2$

Multiply $- 10$ by $0$ .

$y = 0 + 0 - 2$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 2$

Subtract $2$ from $0$ .

$y = - 2$

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{5 - \sqrt{17}}{4}, 0), (- \frac{5 + \sqrt{17}}{4}, 0)$

y-intercept(s): $(0, - 2)$

Step 4

Find the x and y Intercepts f(x)=-4x^2-10x-2

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